Statuary Hall is an elliptical room in the United States Capitol Building in Washington, D.C. The room is also referred to as the Whispering Gallery because a person standing at one focus of the room can hear even a whisper spoken by a person standing at the other focus. A student takes measurements of the room and estimates that the dimensions of Statuary Hall are 47 feet wide by 94 feet long.
(a) Find an equation for the shape of the floor surface of the hall. (Assume the major axis is horizontal and the center of the room is at the origin.)
(b) Determine the distance between the foci. (Round your answer to one decimal place.)
a = 94/2 = 47
b = 47/2 = 23.5
x^2/47^2 + y^2/23.5^2 = 1
now google ellipse to find focus :)
for the focus, c^2=a^2-b^2
The distance between the foci is 2c
"the dimensions of Statuary Hall are 47 feet wide by 94 feet long." That's actually wrong. Instead, "Statuary Hall is 92 ft. wide and 97 ft. long." The hall is only half an ellipse as a set of columns cuts off the other (non-existing) half of the ellipse. Look at floor plans and you can see that the hall is almost circular.
To find the equation for the shape of the floor surface of Statuary Hall, we can use the formula for the equation of an ellipse centered at the origin:
x^2/a^2 + y^2/b^2 = 1
where 'a' is the semi-major axis and 'b' is the semi-minor axis.
In this case, the width of the hall is equal to 2a, so a = 47 / 2 = 23.5 feet. The length of the hall is equal to 2b, so b = 94 / 2 = 47 feet.
Plugging these values into the equation, we have:
x^2 / (23.5^2) + y^2 / (47^2) = 1
Simplifying this equation will give us the final result for part (a).
To determine the distance between the foci, we can use the formula:
c = sqrt(a^2 - b^2)
where 'c' is the distance between the foci.
Plugging in the values of 'a' and 'b' obtained previously:
c = sqrt((23.5^2) - (47^2))
Evaluating this expression will give us the distance between the foci.